We discussed administrative matters and introduced ourselves. Then we began trying to understand which functions satisfy the property f(x+y) = f(x) + f(y). Click below for a more detailed lecture summary.

Click here for Lecture 1 notes

We continued trying to `solve' the equation f(x+y) = f(x) + f(y) for the function f. We made some progress, in particular showing that f(a/b) = (a/b) f(1) for any fraction a/b. We also introduced some notation: for the real numbers, for the integers, symbols `for all' and `contained in', QED, etc. Click below for a more detailed lecture summary.

Click here for Lecture 2 notes

Today we discussed rational versus irrational numbers. In particular, we proved that the square-root of 2 must be irrational. Click below for a more detailed lecture summary.

Click here for Lecture 3 notes

Today we went back to discussing functions satisfying
f(x+y) = f(x) + f(y). We discovered that if f is *continuous*,
then our conjecture that f(x)=x f(1) for every real number x is true!
But what if our function is not continuous? Do such f even exist?
The answer to the latter question depends on whether or not you
believe the Axiom of Choice. Click below for a more detailed summary.

Click here for Lecture 4 notes

Today we began exploring linear maps, the object of interest
in linear algebra. We played around with two simple types of linear
maps -- those from **R** to **R** and those from **R**^{2}
to **R** -- and were able to completely characterize them.

Click here for Lecture 5 notes

Today we saw that our characterizations of linear maps from **R**
to **R** and from **R ^{2}** to

Click here for Lecture 6 notes

We saw two ways to approach a problem about linear maps
from **R ^{2}** to

Click here for Lecture 7 notes

Today we explored the rotation map. We came up with a geometric explanation of why this map is linear. We also used polar coordinates to discover a formula for how the map acts on a given point (x,y).

Click here for Lecture 8 notes

We rephrased the action of a linear map from
**R ^{2}** to

Click here for Lecture 9 notes

We looked at how different linear maps act geometrically on the plane. We also defined the notions of composition and image.

Click here for Lecture 10 notes

We proved that the image of a linear map is either the entire plane (in which case it's called nonsingular), or else is entirely contained inside of some line (in which case the map is called singular). We also discussed the notion of invertibility, and stated a theorem about the invertibility of nonsingular linear maps.

Click here for Lecture 11 notes

Today we proved that every nonsingular linear map is invertible, and asserted that its inverse is also linear. We also proved that the composition of any two linear maps is linear, and established a formula for the matrix of the composition of two maps in terms of the matrices of those two individually.

Click here for Lecture 12 notes

We've discussed how linear maps affect shapes. Today we looked at how they affect area. This led us to define the determinant of a map.

Click here for Lecture 13 notes

We first verified our conjecture from last time about the determinant of a composition. We then gave an application of this to prove that rectangles scale nicely under linear maps. We concluded by pointing out that linear maps, although defined on points, are in fact well-defined on vectors.

Click here for Lecture 14 notes

Today we continued discussing vectors, in particular
proving that any linear map f from **R**^{2}
to **R**^{2}, when viewed as a function on
the set V^{2} of all vectors in the plane, remains
well-defined. We ended by hinting at change of basis.

Click here for Lecture 15 notes

After reviewing our work from last time, we resumed our exploration of change of basis. We realized that there was an implicit linear map at play, which we can think of as an English-Shibbolese dictionary. If this map is nonsingular, we proved that any vector can be expressed in Shibbolese.

Click here for Lecture 16 notes

We concluded our discussion of change-of-basis. We then turned to the Singular Value Decomposition. We considered it from two perspectives: one as a method of understanding what a map f does geometrically, the other as a way of thinking about f as mapping a rectangular lattice to another rectangular lattice.

Click here for Lecture 17-18 notes

We completed our discussion of the singular value decomposition, proving various unproved assertions from last time. We then began exploring powers of a certain matrix, which turned out to be related to the Fibonacci numbers.

Click here for Lecture 19-20 notes

We first discussed associativity, specifically of function composition. Next we proved (by induction) a formula for powers of a certain matrix in terms of Fibonacci numbers. Finally, we motivated and defined the notion of similarity, and outlined our strategy for finding a formula for Fibonacci numbers.

Click here for Lecture 21 notes

We found a formula for the n-th Fibonacci number using matrices. This inspired us to study the spectral decomposition, which in turn led us to explore eigenvalues and eigenvectors.

Click here for Lecture 22--24 notes

We introduced the notion of vector space, and
explored a number of examples. The material is very
similar to Chapter 1, Section 1 of the textbook.
(The book,
Linear Algebra Done Wrong by Sergei Treuil,
is available for free (and legal!) pdf download.) Perhaps the biggest
difference from the book was the addition of one special example:
MSS_{3}, the vector space of all 3x3 magic squares.

We warmed up by proving that 0**v** = **0** for any
vector **v** in a vector space V. We then defined the notion of
a *basis* of a vector space, and gave examples and nonexamples
of bases. See section 2 of the textbook.

We warmed up by proving that a vector space has a unique additive
identity (which we call **0**). We then defined the concepts of
spanning set, linear dependence, and linear independence. Finally,
we proved that a set of vectors is linearly independent if and only if
none of the vectors can be written as a linear combination of the others.
See section 2 of the textbook.

We proved what I called the Fundamental Property of Bases (Treuil
calls it by the less romantic name Proposition 2.7). Then we proved
that any two bases of a vector space V have the same number of elements.
(We did this by using the Steinitz Exchange Trick.) This allowed us
to define the *dimension* of a vector space: it's the number of
vectors in a basis. (See the notes posted below for the proof that every
basis has the same number of elements.)

Click here for some notes on lecture 28

We discussed how to use the fundamental property of bases to determine
whether or not a given set of vectors is a basis. In particular, we proved
that {(1,1),(1,-1)} is a basis of **R**^{2}, while
{(1,2,-1),(2,-1,1),(8,1,1)} is *not* a basis of **R**^{3}.
We then proved that any spanning set contains a basis (Proposition 2.8 in
Chapter 1 of the textbook).

We began by defining what it means for a vector space to be
finite-dimensional: that there exists a finite spanning set. This
definition immediately allowed us to prove that any finite-dimensional
vector space has a basis. Next, we sketched a proof that any linearly
independent set of vectors in a finite-dimensional space is contained
in a basis of that space; you will write out this proof rigorously in
this week's Problem Set. Finally, we defined the notion of a linear map
from one vector space to another (see Chapter 1, Section 3 of the text;
note that Treil calls this a linear transformation). We concluded by
considering four examples of linear maps. The first was the function
f : **R**^{3} → **R**^{2} defined by
f(x,y,z) = (x,y). The second was g : **R**^{2} →
**R**^{3} defined by g(x,y) = (x,y,2x+y). The third example
was the differential operator: d/dx : **P**_{n} →
**P**_{n-1}. The final example we considered was
h : MSS_{3} → **R** which sends a magic square to its
magic sum.

Today we discussed the concepts of invertibility and isomorphisms. Since the definition I gave is different from that in the book, I've written up a lecture summary (click below).

Click here for notes on lecture 31

Today we introduced notions which will help us measure
how far a linear map is from being an isomorphism: these
are the *image* and *kernel* of the map. Along the
way we defined the notion of subspace, and stated
the Rank-Nullity Theorem.

Click here for notes on lecture 32

We started by proving the Rank-Nullity Theorem. Then we explored some of its consequences.

Click here for notes on lecture 33

We discussed how to represent an abstract linear map as a matrix. Among other things, we represented the differential operator as a matrix.

Click here for notes on lecture 34

First, we discussed how to view abstract vector spaces (and associated linear maps) in a more concrete way. Next, we talked about the determinant of a linear map; we went over two methods (the notes also contain a description of another method due to Lewis Carroll, but that's just for fun -- you won't be tested on that, of course). Our second approach led to the concept of triangular matrices, and also to a method for finding the inverse of a given matrix. Finally, we briefly touched on the singular value decomposition and the spectral decomposition.